A028391 a(n) = n - floor(sqrt(n)).

0, 0, 1, 2, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65

Offset: 0

Author

John Mellor (u15630(AT)snet.net)

Keywords

• nonn
• easy
• nice

Comments

Number of nonsquares <= n.

Number of k <= n with an even number of divisors. - Benoit Cloitre, Sep 07 2002

Construct the pyramid

............a(0)

.......a(1).a(2).a(3)

..a(4).a(5).a(6).a(7).a(8).. etc.

Now circle all the primes and the result will be a pattern very similar to the famous Ulam spiral. - Sam Alexander, Nov 14 2003

The sequence floor(n-n^(1/2)) gives the same numbers with a different offset. - Mohammad K. Azarian, R. J. Mathar and M. F. Hasler, Apr 30 2008

The number of nonzero values of floor (j^2/n) taken over 1 <= j <= n-1.

a(n) = A173517(n) iff n is not a square. - Reinhard Zumkeller, Feb 20 2010

a(n) - a(n-1) = 0 if n is a square, otherwise 1. - Robert Israel, Dec 30 2014

References

• B. Alspach, K. Heinrich and G. Liu, Orthogonal factorizations of graphs, pp. 13-40 of Contemporary Design Theory, ed. J. H. Dinizt and D. R. Stinson, Wiley, 1992 (see Theorem 2.7).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Dick Boland, Introduction to the Square Spine Spiral, 2000-2003 [broken link].

Crossrefs

Cf. A056847, A000196, A135662-A135665, A166373.

Programs

Haskell

a028391 n = n - a000196 n  -- Reinhard Zumkeller, Oct 28 2012

Magma

[n-Floor(Sqrt(n)): n in [0..100]]; // Vincenzo Librandi Dec 31 2014

Maple

seq(n - floor(sqrt(n)), n = 0 .. 100); # Robert Israel, Dec 30 2014

Mathematica

f[n_]:=n-Floor[Sqrt[n]];Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2010 *)

PARI

a(n)=n-sqrtint(n) \\ Charles R Greathouse IV, Jun 28 2013

Python

from math import isqrt
def A028391(n): return n-isqrt(n) # Chai Wah Wu, Jul 28 2022

Formula

a(n) = ceiling(n - sqrt(n)), as follows from ceiling(-x) = -floor(x). [Corrected by M. F. Hasler, Feb 21 2010]

a(n) = 2*n - A028392(n). - Reinhard Zumkeller, Oct 28 2012

G.f.: (1+x)/(2*(1-x)^2) - Theta3(0,x)/(2*(1-x)) where Theta3 is a Jacobi theta function. - Robert Israel, Dec 30 2014

Extensions

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, May 01 2008

Comment and cross-reference added by Christopher Hunt Gribble, Oct 13 2009

Formula corrected by M. F. Hasler, Feb 21 2010

More terms from Vladimir Joseph Stephan Orlovsky, Mar 29 2010